# Torsion in tensor products over one-dimensional domains

## TL;DR

This paper investigates conditions under which the tensor product of certain modules over a one-dimensional Gorenstein local domain remains torsion-free, revealing structural constraints on the endomorphism ring.

## Contribution

It establishes that if such modules exist with torsion-free tensor product, then the endomorphism ring must be local with the same residue field as the base domain.

## Key findings

- Existence of non-free modules with torsion-free tensor product implies the endomorphism ring is local.
- The endomorphism ring shares the same residue field as the original domain.
- Provides structural insight into modules over one-dimensional Gorenstein domains.

## Abstract

Over a one-dimensional Gorenstein local domain $R$, let $E$ be the endomorphism ring of the maximal of $R$, viewed as a subring of the integral closure $\overline R$. If there exist finitely generated $R$-modules $M$ and $N$, neither of them free, whose tensor product is torsion-free, we show that $E$ must be local with the same residue field as $R$.